# Section1.2Vector Equations and Spans¶ permalink

##### Objectives
1. Understand the equivalence between a system of linear equations and a vector equation.
2. Learn the definition of and how to draw pictures of spans.
3. Pictures: an inconsistent system of equations, a consistent system of equations, spans in and
4. Vocabulary: vector equation.
5. Essential vocabulary: span.

# Subsection1.2.1Vector Equations

An equation involving vectors with entries is the same as equations involving only numbers. For example, the equation

(1.2.1)

simplifies to

For two vectors to be equal, all of their coordinates must be equal, so this is just the system of linear equations

##### Definition

A vector equation is an equation involving a linear combination of vectors with possibly unknown coefficients.

Asking whether or not a vector equation has a solution is the same as asking if a given vector is a linear combination of some other given vectors.

For example the vector equation above is asking if the vector is a linear combination of the vectors and

##### Example

Consider the vector equation

Does this vector equation have a solution? Does it have more than one solution?

This vector equation does not have any solution. We could write it as three separate equations:

which all have to be true at the same time for a solution The second equation says and the third says These two statements cannot be true at the same time, no matter what values of and we choose. So there can be no solution to the vector equation.

##### Definition

If one or more solutions exist for an equation or a system of equations, it is said to be consistent. If an equation or system of equations does not have any solution, it is said to be inconsistent.

The above definition is the first of several essential definitions that we will see in this textbook. They are essential in that they form the essence of the subject of linear algebra: learning linear algebra means (in part) learning these definitions. All of the definitions are important, but it is essential that you learn and understand the definitions marked as such.

##### A Picture of a Consistent System

Below we will show that the above system of equations is consistent. Equivalently, this means that the above vector equation has a solution. In other words, there is a linear combination of and that equals We can visualize the last statement geometrically. Therefore, the following figure gives a picture of a consistent system of equations. Compare with figure below, which shows a picture of an inconsistent system.

# Subsection1.2.2Spans

It will be useful to know what are all linear combinations of a set of vectors in In other words, we would like to understand the set of all vectors in such that the vector equation (in the unknowns )

has a solution (i.e. is consistent).

##### Definition

Let be vectors in The span of is the collection of all linear combinations of and is denoted

In symbols:

We also say that is the subset spanned by or generated by the vectors

##### Set Builder Notation

as: “the set of all things of the form such that are in The vertical line is “such that”; everything to the left of it is “the set of all things of this form”, and everything to the right is the condition that those things must satisfy to be in the set. Specifying a set in this way is called set builder notation.

All mathematical notation is only shorthand: any sequence of symbols must translate into an ordinary sentence.

##### Note(Consistency and span)

Here are two ways of saying the same thing:

1. A vector is in the span of
2. The vector equation
is consistent, i.e., it has at least one solution.

Later, in subsection in Section 2.2 we will develop a procedure for answering the question “is in the span of in every circumstance. This will be a byproduct of developing a process to find all the solutions of vector equations.

##### Pictures of Spans

Drawing a picture of is the same as drawing a picture of all linear combinations of